The two consecutive odd positive integers can be written as: 2x + 1, 2x + 3, where x is integer and x>=0.

The sum of the squares of 2x + 1 and 2x + 3 is 130 means:

(2x + 1)^2 + (2x + 3)^2 = 130

........

click here to see the equation solved for x

........

x = 3 

We need to ignore the negative solution -5 because x can only be positive.

2*3 + 1 = 7

2*3 + 3 = 9

Indeed: 

7^2 + 9^2 = 49 + 81 = 130 

The two consecutive odd positive integers are 7 and 9.